186 XIII INTEGRATION
End of the Proof
(13.14.1.7) For every function veS such that/^ v, we have^ g vg, and
f* v dv = I* vg dp by (13.14.1.2), so that $* fg dp £ J* vg d^ = J* v dv.
Hence, by definition of the upper integral, we have J* fg d\i g J*/^v, Hence
it remains to establish the opposite inequality
(13.14.1.8) |/dvg \*fgdfj..
Let h e«/ be such that h^fg. Then it is enough to show that
(13.14.1.9) \*fdv£ hdiJL.
The set X — A is the union of a denumerable increasing sequence of
compact sets Hn and a ju-negligible set N, such that g \ Hn is continuous, finite
and >0 for all n. We define a mapping u of X into R as follows: u = h/g
in the union of the sets Hn, and u(x) = + oo in N and in A. In each of the sets
Hn we have ug = h, and by virtue of (13.14.1.6)
r* r**
u dv = w
JHn jHn
But v(N) = 0 by virtue of (13.14.1.3), and v(A) = 0 by virtue of (13.14.1.5),
hence, as h ^ 0, (13.5.7)
r* p* r* r*
u dv = sup u dv = sup /i dju g /i djU.
J n JHM n jHn J
Since/^w, we obtain the required inequality (13.14.1.9). Q.E.D.
(13.14.2) Let g be a locally ii-integrable function 'which is ^0 on X, /ef S be
the ^-measurable set of points xeX such that g(x) > 0, and let v = g • /i. Le^
/6e a mapping ofX into R. Then the following conditions (with the conventions
of (13.11) for products) are equivalent:
(a) / is v-measurable ;
(b) /<ps w \n-measurable;
(c) y<7 z,y \i-measurable.
We may suppose # to be finite. With the conventions we have made, we
have jfc = (f(psK09s)> and since the two factors on the right-hand side neverp, = \ fg d^